\section{Conclusion}
\label{sec:conclusion}

In this paper, we have shown that it is simple to characterize a critical instant for sporadic tasks with self-suspensions, thereby invalidating a claim made in earlier works. We highlighted the complexity of the problem and presented an algorithm to compute the exact WCRT of a lower priority self-suspending tasks when scheduled together with non-self-suspending sporadic tasks. As the algorithm rapidly becomes intractable for a large number of higher priority-tasks due to the exponential number of scenarios that need to be considered, we formulated a response time test for multiple suspension regions as an optimization problem that can be solved by a MILP tool in reasonable time. The optimization problem was then extended to accommodate multiple self-suspending sporadic tasks interfering with each other, still under fixed-priority scheduling. Experiment results showed that the proposed response time tests dominate state-of-the-art techniques, % that are correct in a sufficient sense, 
although the WCRT gains highly depend on the peculiarities of the task sets. Experiments also pointed out that the optimization problem finds the exact WCRT solution in the majority of the cases.